Definition 39.4.1. Let $S$ be a scheme.

1. A group scheme over $S$ is a pair $(G, m)$, where $G$ is a scheme over $S$ and $m : G \times _ S G \to G$ is a morphism of schemes over $S$ with the following property: For every scheme $T$ over $S$ the pair $(G(T), m)$ is a group.

2. A morphism $\psi : (G, m) \to (G', m')$ of group schemes over $S$ is a morphism $\psi : G \to G'$ of schemes over $S$ such that for every $T/S$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).